Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C3 |
No aliases for this group.
| Fitting | C3×Q8×C9 ⋊ C2 |
| Sylow | C3×C9 ⋊ #41 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×C3⋊D9 |
| commutator, cosocle | C2×C3×C9 | C23 |
| Frattini | C2×C3 | C22×C3⋊D3 |
| Fitting | C3×Q8×C9 | C2 |
| socle | C2×C32 | C22×D3 |
| hypercenter | Q8 | C3⋊D9 |
| hypocenter | C3×C9 | #41 |
| LCS | G ▷ C2×C3×C9 ▷ C3×C9 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
200 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 216 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,3)(13,15,16,19)(14,17,18,20);(4,9,11,10,5,7,6,12,8)(13,14,16,18)(15,20,19,17);(2,3)(5,7)(6,10)(8,9)(11,12)(15,19)(17,20)Polycyclic presentation:
2^4,3^3|b^2=d,c^2=d,e^3=g^2,c^a=cd,e^a=e^2g,f^a=f^2,g^a=g^2,c^b=cd75 conjugacy classes — view the full character table.
| maximal subgroup of | #11721213, #11721216, #11721221, … |
| maximal normal subgroup of | #11721213, #11721216, #11721221, … |
| maximal quotient of | #11721746, #11721748, #11721749, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 393);